TITLE: Unconditionally convergent series in $\ell_2$ consisting of $\ell_1$-small vectors
QUESTION [6 upvotes]: For a function $x:\omega\to\mathbb R$ let $|x|$ denote the function $|x|:\omega\to[0,\infty)$, $|x|:n\mapsto|x(n)|$.
It is well-know that a series $\sum_{n\in\omega}r_n$ of real numbers converges unconditionally if and only if $\sum_{n\in\omega}|r_n|<\infty$.
On the other hand, there exists an unconditionally convergent series $\sum_{n\in\omega}x_n$ in $\ell_2$ such that the series $\sum_{n\in\omega}|x_n|$ is divergent.
I am interested in finding conditions on an unconditionally convergent series $\sum_{n\in\omega}x_n$ in $\ell_2$ guaranteeing that the series $\sum_{n\in\omega}|x_n|$ converges.

Question. Let $I:\ell_1\to\ell_2$ be the identity operator and $(x_n)_{n\in\omega}$ be a sequence of elements of $\ell_1$ such that $\sum_{n\in\omega}\|x_n\|^2<\infty$ and the series $\sum_{n\in\omega}I(x_n)$ unconditionally converges in $\ell_2$. Is the series $\sum_{n\in\omega}|I(x_n)|$ convergent in $\ell_2$?

REPLY [6 votes]: If I understand correctly what you are asking, the answer is "certainly not".
Consider $k$ orthogonal vectors $v_j$ with $k$ coordinates $\pm 1$ (the Hadamard matrix). Now multiply them by $t$ and take $n$ such identical bunches. Then the sum of $\ell^1$ norms squared is $nt^2k^3$, the squared $\ell^2$ norm of the sum of absolute value vectors is $n^2t^2k^3$ and the maximum of the squared $\ell^2$ norms of subset sums is $n^2t^2k^2$. Then playing this game on disjoint bunches of coordinates, we see that the question is whether the conditions $\sum_m n_mt_m^2k_m^3<+\infty$ and $\sum_m n_m^2t_m^2k_m^2<+\infty$ imply $\sum_m n_m^2t_m^2k_m^3<+\infty$. Now just take $n_m=k_m=2^m$ and $t_m=2^{-2m}/m$, say.
But perhaps you meant something else? :-)